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solve -- solve a linear equation

Synopsis

Description

(Disambiguation: for division of matrices, which can also be thought of as solving a system of linear equations, see instead Matrix // Matrix. For lifting a map between modules to a map between their free resolutions, see extend.)

There are several restrictions. The first is that there are only a limited number of rings for which this function is implemented. Second, over RR or CC, the matrix A must be a square non-singular matrix. Third, if A and b are mutable matrices over RR or CC, they must be dense matrices.
i1 : kk = ZZ/101;
i2 : A = matrix"1,2,3,4;1,3,6,10;19,7,11,13" ** kk

o2 = | 1  2 3  4  |
     | 1  3 6  10 |
     | 19 7 11 13 |

              3        4
o2 : Matrix kk  <--- kk
i3 : b = matrix"1;1;1" ** kk

o3 = | 1 |
     | 1 |
     | 1 |

              3        1
o3 : Matrix kk  <--- kk
i4 : x = solve(A,b)

o4 = | 2  |
     | -1 |
     | 34 |
     | 0  |

              4        1
o4 : Matrix kk  <--- kk
i5 : A*x-b

o5 = 0

              3        1
o5 : Matrix kk  <--- kk
Over RR or CC, the matrix A must be a non-singular square matrix.
i6 : printingPrecision = 2;
i7 : A = matrix "1,2,3;1,3,6;19,7,11" ** RR

o7 = | 1  2 3  |
     | 1  3 6  |
     | 19 7 11 |

                3          3
o7 : Matrix RR    <--- RR
              53         53
i8 : b = matrix "1;1;1" ** RR

o8 = | 1 |
     | 1 |
     | 1 |

                3          1
o8 : Matrix RR    <--- RR
              53         53
i9 : x = solve(A,b)

o9 = | -.15 |
     | 1.1  |
     | -.38 |

                3          1
o9 : Matrix RR    <--- RR
              53         53
i10 : A*x-b

o10 = | 0        |
      | -3.3e-16 |
      | -8.9e-16 |

                 3          1
o10 : Matrix RR    <--- RR
               53         53
i11 : norm oo

o11 = 8.88178419700125e-16

o11 : RR (of precision 53)
For large dense matrices over RR or CC, this function calls the lapack routines.
i12 : n = 10;
i13 : A = random(CC^n,CC^n)

o13 = | .92+.57i .37+.4i  .93+.84i  .53+.09i  .51+.09i .76+.1i   .43+.25i 
      | .38+.86i .29+.84i .18+.6i   .77+.53i  .49+.44i .8+.57i   .45+.053i
      | .68+.62i .24+.42i .14+.072i .44+.063i .11+.96i .46+.2i   .4+.71i  
      | .26+.31i .78+.63i .38+.043i .094+.43i .83+.55i .71+.86i  .04+.46i 
      | .22+i    .59+.21i .96+.37i  .29+.67i  .36+.75i .46+.66i  .67+.21i 
      | .27+.89i .45+.93i .86+.17i  .91+.92i  .78+.69i .94+.08i  .78+.16i 
      | .86+.56i .75+.16i .56+.65i  .32+.67i  .13+.21i .081+.49i .73+.77i 
      | .11+.74i .96+.35i .58+.87i  .66+.98i  .14+.27i .7+.42i   .2+.47i  
      | .58+.92i .3+.52i  .33+i     .87+.29i  .91+.49i .74+.28i  .81+.09i 
      | .17+.2i  .1+.94i  .47+.56i  .01+.54i  .43+.89i .39+.89i  .06+.64i 
      -----------------------------------------------------------------------
      .07+.85i   .75+.31i  .068+.11i |
      .85+.87i   .46+.071i .07+.86i  |
      .91+.72i   .8+.02i   .93+.56i  |
      .35+.22i   .81+.1i   .06+.76i  |
      .34+.36i   .78+.15i  .23+.24i  |
      .63+.63i   .4+.92i   .65+.9i   |
      .011+.054i .64+.49i  .15+.45i  |
      .49+.88i   .8+.97i   .25+.49i  |
      .95+.12i   .54+.15i  .63+.25i  |
      .09+.81i   .33+.035i .97+.46i  |

                 10          10
o13 : Matrix CC     <--- CC
               53          53
i14 : b = random(CC^n,CC^2)

o14 = | .21+.8i  .66+.1i   |
      | .01+.63i .2+.47i   |
      | .79+.91i .91+.14i  |
      | .92+.14i .24+.67i  |
      | .24+.58i .98+.16i  |
      | .58+.51i .03+.78i  |
      | .4+.8i   .096+.23i |
      | .54+.68i .43+.65i  |
      | .24+.85i .87+.42i  |
      | .85+.81i .8+.47i   |

                 10          2
o14 : Matrix CC     <--- CC
               53          53
i15 : x = solve(A,b)

o15 = | .63+.86i   -.04-.91i |
      | -.9-.98i   -.68+.68i |
      | -.013+.44i .34+.059i |
      | -.54-1.1i  -1.2+.52i |
      | -1.1+.23i  .14+1.1i  |
      | 1.5-.26i   .31-i     |
      | -.22-.036i .078+.22i |
      | .47+1.1i   .63-.29i  |
      | 1.3-.35i   .51-.41i  |
      | .35+.81i   1.2+.2i   |

                 10          2
o15 : Matrix CC     <--- CC
               53          53
i16 : norm ( matrix A * matrix x - matrix b )

o16 = 9.42055475210265e-16

o16 : RR (of precision 53)
This may be used to invert a matrix over ZZ/p, RR or QQ.
i17 : A = random(RR^5, RR^5)

o17 = | .45 .26 .57  .55 .94 |
      | .38 .98 .38  .45 .99 |
      | .8  .6  .093 .84 .4  |
      | .39 .2  .76  .51 .46 |
      | .16 .84 .61  .78 .31 |

                 5          5
o17 : Matrix RR    <--- RR
               53         53
i18 : I = id_(target A)

o18 = | 1 0 0 0 0 |
      | 0 1 0 0 0 |
      | 0 0 1 0 0 |
      | 0 0 0 1 0 |
      | 0 0 0 0 1 |

                 5          5
o18 : Matrix RR    <--- RR
               53         53
i19 : A' = solve(A,I)

o19 = | -2   .96  1.1  2.4  -1.9  |
      | -1.7 1.3  .029 .57  .023  |
      | -1.2 .4   -.59 2.3  -.23  |
      | 2.4  -1.9 .36  -2.3 1.8   |
      | 1.8  .065 -.4  -1.3 -.032 |

                 5          5
o19 : Matrix RR    <--- RR
               53         53
i20 : norm(A*A' - I)

o20 = 4.44089209850063e-16

o20 : RR (of precision 53)
i21 : norm(A'*A - I)

o21 = 4.44089209850063e-16

o21 : RR (of precision 53)
Another method, which isn't generally as fast, and isn't as stable over RR or CC, is to lift the matrix b along the matrix A (see Matrix // Matrix).
i22 : A'' = I // A

o22 = | -2   .96  1.1  2.4  -1.9  |
      | -1.7 1.3  .029 .57  .023  |
      | -1.2 .4   -.59 2.3  -.23  |
      | 2.4  -1.9 .36  -2.3 1.8   |
      | 1.8  .065 -.4  -1.3 -.032 |

                 5          5
o22 : Matrix RR    <--- RR
               53         53
i23 : norm(A' - A'')

o23 = 0

o23 : RR (of precision 53)

Caveat

This function is limited in scope, but is sometimes useful for very large matrices

See also

Ways to use solve :